Group testing is an approach aimed at identifying up to d defective items among a total of n elements. This is accomplished by examining subsets to determine if at least one defective item is present. We focus on the problem of identifying a subset of \(\ell < d\) defective items. We develop upper and lower bounds on the number of tests required to detect \(\ell \) defective items in both the adaptive and non-adaptive settings while considering scenarios where no prior knowledge of d is available, and situations where some non-trivial estimate of d is at hand. When d is unknown, we prove a lower bound of \( \varOmega (\frac{\ell \log ^2n}{\log \ell +\log \log n})\) tests in the randomized non-adaptive settings and an upper bound of \(O(\ell \log ^2 n)\) for the same settings. Furthermore, we demonstrate that any non-adaptive deterministic algorithm must make \(\varTheta (n)\) tests, signifying a fundamental limitation in this scenario. For adaptive algorithms, we establish tight bounds in different scenarios. In the deterministic case, we prove a tight bound of \(\varTheta (\ell \log {(n/\ell )})\) . Moreover, in the randomized settings, we derive a tight bound of \(\varTheta (\ell \log {(n/d)})\) . When d, or at least some non-trivial estimate of d, is known, we prove a tight bound of \(\varTheta (d\log (n/d))\) for the deterministic non-adaptive settings, and \(\varTheta (\ell \log (n/d))\) for the randomized non-adaptive settings. In the adaptive case, we present an upper bound of \(O(\ell \log (n/\ell ))\) for the deterministic settings, and a lower bound of \(\varOmega (\ell \log (n/d)+\log n)\) . Additionally, we establish a tight bound of \(\varTheta (\ell \log (n/d))\) for the randomized adaptive settings.
